Prologue Some ideas about strange attractors

Abstract

For two-dimensional diffeomorphisms or flows reducing essentially to them the evolution of S.A. can be described geometrically using bifurcations, homoclinic and heteroclinic points. However, many questions are left open:

1)

Prediction of values of the parameters for which a S.A. appears or is suddenly destroyed.

2)

Existence of invariant measures on the S.A. Ergodic or mixing properties of the diffeomorphism restricted to the S.A., with respect to this measure.

3)

Examination of the geometry of the S.A. for higher dimensions. Mechanisms producing or destroying S.A. in this case: Study of homo/heteroclinic points of normally hyperbolic invariant or periodic objects.

We strongly recommend to look for the geometric structure in physical or numerical experiments. It seems to us that without this knowledge one cannot get a really deep insight in the problem of S.A.